Some results for the Jacobi-Dunkl transform in the space
L
2(
R
,
A
α,β
(t)dt)
S. EL OUADIH
a,∗, R. DAHER
band A. BELKHADIR
ca,b,cDepartement of Mathematics, Faculty of Sciences A¨ın Chock,University Hassan II, Casablanca, Morocco.
Abstract
In this paper, using a generalized Jacobi-Dunkl translation operator, we prove an analog of Titchmarsh’s theorem for functions satisfying the Jacobi-Dunkl Lipschitz condition in L2(R,Aα,β(t)dt),α ≥ β ≥ −21,α6=
−1
2 .
Keywords: Jacobi-Dunkl operator, Jacobi-Dunkl transform, generalized Jacobi-Dunkl translation.
2010 MSC:42B37. c2012 MJM. All rights reserved.
1
Introduction
Titchmarsh’s [[10], Theorem 85] characterized the set of functions in L2(R)satisfying the Cauchy-Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transform, namely we have: Theorem 1.1. [10] Letα∈(0, 1)and assume that f ∈L2(R). Then the following statements are equivalents:
(a) kf(t+h)−f(t)k=0(hα), as h→0,
(b)
Z
|λ|≥r
|fb(λ)|2dλ=O(r−2α), as r→∞,
where bf stands for the Fourier transform of f .
In this paper, we prove in analog of Theorem 1.1 for the Jacobi-Dunkl transform for functions satisfying the Jacobi-Dunkl Lipschitz condition in the space L2(R,Aα,β(t)dt). For this purpose, we use the generalized translation operator. Similar results have been established in the context of non compact rank one Riemannian symetric spaces [9].
In section 2 below, we recapitulate from [[1],[2],[3],[5]] some results related to the harmonic analysis associated with Jacobi-Dunkl operatorΛα,β. Section 3 is devoted to the main result after defining the classLip(δ, 2,α,β)
of functions in L2α,β(R) satisfying the Lipschitz condition correspondent to the generalized Jacobi-Dunkl translation.
2
Notation and Preliminaries
The Jacobi-Dunkl function with parameters(α,β),α≥β≥ −21,α6= −21, is defined by the formula:
∀x∈R,ψα,βλ (x) =
ϕα,βµ (x)−λi dxd ϕα,βµ (x) ifλ∈C\{0},
1 ifλ=0,
withλ2=µ2+ρ2,ρ=α+β+1 andϕα,βµ is the Jacobi function given by:
ϕα,βµ (x) =F
ρ+iµ
2 ,
ρ−iµ
2 ,α+1,−(sinhx) 2,
∗Corresponding author.
where F is the Gausse hypergeometric function (see [1],[6] and [7]).
ψα,βλ is the uniqueC∞-solution onRof the differentiel-difference equation
Λα,βU =iλU, λ∈C,
U(0) =1, whereΛα,βis the Jacobi-Dunkl operator given by:
Λα,βU(x) = d U(x)
dx + [(2α+1)cothx+ (2β+1)tanhx]×
U(x)− U(−x)
2 ].
The operatorΛα,βis a particular case of the operatorDgiven by
DU(x) = dU(x)
dx + A0(x)
A(x) ×
U(
x)− U(−x) 2
,
where A(x) = |x|2α+1B(x), and B a function of class C∞ on R, even and positive. The operator Λ α,β corresponds to the function
A(x) =Aα,β(x) =2ρ(sinh|x|)2α+1(cosh|x|)2β+1. Using the relation
d dxϕ
α,β
µ (x) =−
µ2+ρ2
4(α+1)sinh(2x)ϕ
α+1,β+1 µ (x), the functionψλα,βcan be written in the form below (see [2])
ψα,βλ (x) =ϕα,βµ (x) +i λ
4(α+1)sinh(2x)ϕ
α+1,β+1
µ (x), x ∈R, whereλ2=µ2+ρ2, ρ=α+β+1.
DenoteL2α,β(R) =L2α,β(R,Aα,β(t)dt)the space of measurable functions f onRsuch that
kfkL2
α,β(R)
=
Z
R|f(t)|
2A α,β(t)dt
1/2
<+∞.
Using the eigenfunctionsψα,βλ of the operatorΛα,βcalled the Jacobi-Dunkl kernels, we define the Jacobi-Dunkl transform of a functionf ∈L2
α,β(R)by: Fα,βf(λ) =
Z
R f(t)ψ
α,β
λ (t)Aα,β(t)dt, λ∈R, and the inversion formula
f(t) =
Z
RFα,βf(λ)ψ
α,β
−λ(t)dσ(λ), where
dσ(λ) = |λ|
8πpλ2−ρ2|Cα,β(
p
λ2−ρ2)|
IR\]−ρ,ρ[(λ)dλ.
Here,
Cα,β(µ) = 2
ρ−iµΓ(α+1)Γ(iµ) Γ(1
2(ρ+iµ))Γ(12(α−β+1+iµ))
, µ∈C\(iN),
andIR\]−ρ,ρ[is the characteristic function ofR\]−ρ,ρ[.
DenoteL2
σ(R) =L2(R,dσ(λ)).
The Jacobi-Dunkl transform is a unitary isomorphism fromL2α,β(R)ontoL2σ(R), i.e., kfk:=kfkL2
α,β(R)
=kFα,β(f)kL2
The operator of Jacobi-Dunkl translation is defined by: Txf(y) =
Z
R f(z)dν
α,β
x,y(z), ∀x,y∈R, (2.2)
whereνα,βx,y(z), x,y∈R, are the signed measures given by
dνα,βx,y(z) =
Kα,β(x,y,z)Aα,β(z)dz ifx,y∈R∗,
δx ify=0,
δy ifx =0,
here,δxis the Dirac measure atx. And,
Kα,β(x,y,z) = Mα,β(sinh(|x|)sinh(|y|)sinh(|z|))−2αIIx,y×
Z π
0 ρθ(x,y,z) × (gθ(x,y,z))
α−β−1
+ sin2βθdθ,
Ix,y = [−|x| − |y|,−||x| − |y||]∪[||x| − |y||,|x|+|y|],
ρθ(x,y,z) = 1−σxθ,y,z+σzθ,x,y+σzθ,y,x,
∀z∈R,θ∈[0,π],σxθ,y,z=
coshx+coshy−coshzcosθ
sinhxsinhy , ifxy6=0,
0, ifxy=0,
gθ(x,y,z) =1−cosh
2x−cosh2y−cosh2z+2 coshxcoshycoshzcos
θ,
t+ =
t, ift>0, 0, ift≤0, and,
Mα,β =
2−2ρΓ(α+1)
√
πΓ(α−β)Γ(β+12) ifα>β,
0 ifα=β.
In [2], we have
Fα,β(Thf)(λ) =ψλα,β(h)Fα,β(f)(λ), λ,h∈R. (2.3)
Forα≥ −21, we introduce the Bessel normalized function of the first kind defined by:
jα(z) =Γ(α+1)
∞
∑
n=0(−1)n(z2)2n
n!Γ(n+α+1), z∈C.
Moreover, we see that
lim
z→0
jα(z)−1
z2 6=0, by consequence, there existsC1>0 andη>0 satisfying
|z| ≤η⇒ |jα(z)−1| ≥C1|z|2. (2.4) Lemma 2.1. The following inequalities are valids for Jacobi functionsϕα,βµ (t):
(c) |ϕα,βµ (t)| ≤1,
(d) |1−ϕα,βµ (t)| ≤t2(µ2+ρ2). Proof.(See [8], Lemma 3.1, Lemma 3.2).
Lemma 2.2. Letα≥β≥ −21,α6= −21. Then for|ν| ≤ρ, there exists a positive constant C2such that |1−ϕα,βµ+iν(t)| ≥C2|1−jα(µt)|.
3
Main results
In this section we introduce and prove an analog of Theorem 1.1. Firstly we have to define, for functions in L2α,β(R), the conditions of Cauchy-Lipschitz related to the Jacobi-Dunkl translation operator given in 2.2. Definition 3.1. Letδ∈ (0, 1). A function f ∈ L2α,β(R)is said to be in the Jacobi-Dunkl-Lipschitz class, denoted by
Lip(δ, 2,α,β), if
kNhΛmα,βfk=O(hδ), as h→0,
where Nh=Th+T−h−2I, I is the unit operator in the space L2α,β(R)and m=0, 1, 2, .... Lemma 3.3. For f ∈ L2α,β(R), then
kNhΛmα,βfk2=4
Z
Rλ
2m|
ϕα,βµ (h)−1|2|Fα,βf(λ)|2dσ(λ). Proof.SinceFα,β(Λα,βf)(λ) =iλFα,β(f)(λ), we have
Fα,β(Λmα,βf)(λ) =imλmFα,β(f)(λ), m=0, 1, 2, .... (3.5)
We us formulas 2.3 and 3.5, we conclude that Fα,β(NhΛmα,βf)(λ) =im(ψ
α,β λ (h) +ψ
α,β
λ (−h)−2)λ
mF
α,β(f)(λ).
Since
ψα,βλ (h) =ϕα,βµ (h) +i
λ
4(α+1)sinh(2h)ϕ
α+1,β+1 µ (h),
ψα,βλ (−h) =ϕα,βµ (−h)−i λ
4(α+1)sinh(2h)ϕ
α+1,β+1 µ (−h), andϕα,βµ is even (see [2]), then
Fα,β(NhΛmα,βf)(λ) =2im(ϕ α,β
µ (h)−1)λ
mF
α,β(f)(λ).
Now by Parseval’s identity (formula 2.1), we have the result.
Theorem 3.1. Let f ∈ L2α,β(R). Then the following statements are equivalents: (i) f ∈ Lip(δ, 2,α,β),
(ii)
Z
|λ|≥r
λ2m|Fα,β(f)(λ)|2dσ(λ) =O(r−2δ), as r→∞. Proof.(i)⇒(ii). Assume that f ∈ Lip(δ, 2,α,β), then we have
kNhΛmα,βfk=O(hδ), as h→0. From Lemma 3.3, we have
kNhΛmα,βfk2=4
Z
Rλ
2m|1−
ϕα,βµ (h)| 2|F
α,β(f)(λ)|2dσ(λ).
By 2.4 and Lemma 2.2, we get:
Z
η
2h≤|λ|≤ η h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ)≥C12C22
Z
η
2h≤|λ|≤ η h
|µh|4|λ2mFα,β(f)(λ)|2dσ(λ).
From 2ηh ≤ |λ| ≤ ηhwe have
η
2h
2
− ρ2≤µ2≤ η
h
2
−ρ2
⇒ µ2h2≥ η
2 4 −ρ
Takeh≤ η
3ρ, then we haveµ2h2≥C3=C3(η).
So,
Z
η
2h≤|λ|≤ η h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ)≥C12C22C23
Z
η
2h≤|λ|≤ η h
λ2m|Fα,β(f)(λ)|2dσ(λ).
There exists then a positive constantCsuch that
Z
η
2h≤|λ|≤ η h
λ2m|Fα,β(f)(λ)|2dσ(λ)≤C
Z
Rλ
2m|1−
ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ)≤Ch2δ.
For all 0<h< η
3ρ, then we have
Z
r≤|λ|≤2r
λ2m|Fα,β(f)(λ)|2dσ(λ)≤Cr−2δ, r→∞.
Furthermore, we obtain
Z
|λ|≥r
λ2m|Fα,β(f)(λ)|2dσ(λ) =
∞
∑
i=0Z
2ir≤|λ|≤2i+1rλ 2m|F
α,β(f)(λ)|2dσ(λ)
≤ C
∞
∑
i=0(2ir)−2δ ≤ Cr−2δ. This proves that
Z
|λ|≥r
λ2m|Fα,β(f)(λ)|2dσ(λ) =O(r−2δ), as r→∞.
(ii)⇒(i). Suppose now that
Z
|λ|≥rλ
2m|F
α,β(f)(λ)|2dσ(λ) =O(r−2δ), as r→∞,
and write
Z
Rλ
2m|1−
ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ) =
Z
|λ|<1h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ)
+
Z
|λ|≥1h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ).
Using the inequality (c) of Lemma 2.1, we get
Z
|λ|≥1h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ)≤4
Z
|λ|≥1h
λ2m|Fα,β(f)(λ)|2dσ(λ).
Then
Z
|λ|≥1h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ) =O(h2δ), as h→0. (3.6)
Set
φ(λ) =
Z ∞
λ
x2m|Fα,β(f)(x)|2d
σ(x).
An integration by parts gives:
Z x
0 λ 2
λ2m|Fα,β(f)(λ)|2dσ(λ) =
Z x
0 −λ 2
φ0(λ)dλ
= −x2φ(x) +2
Z x
0 λφ(λ)dλ ≤ 2
Z x
0 O
(λ1−2δ)dλ
From Lemma 2.1, we get
Z
|λ|≤1h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ) ≤
Z
|λ|≤1h
λ2m|1−ϕα,βµ (h)||Fα,β(f)(λ)|2dσ(λ)
≤
Z
|λ|≤1h
λ2m(µ2+ρ2)h2|Fα,β(f)(λ)|2dσ(λ)
≤ h2
Z
|λ|≤1h
λ2λ2m|Fα,β(f)(λ)|2dσ(λ)
= O(h2h−2+2δ). Hence,
Z
|λ|≤1h
λ2m|1−ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ) =O(h2δ). (3.7)
Finally, we conclude from 3.6 and 3.7 that
Z
Rλ
2m|1−
ϕα,βµ (h)|2|Fα,β(f)(λ)|2dσ(λ) =
Z
|λ|<1h +
Z
|λ|≥1h = O(h2δ) +O(h2δ) = O(h2δ).
And this ends the proof.
Corollary 3.1. Let f ∈L2α,β(R), and let
kNhΛmα,βfk=O(hδ), as h→0,
Then Z
|λ|≥r
|Fα,β(f)(λ)|2dσ(λ) =O(r−2m−2δ), as r→∞.
4
Acknowledgment
The authors would like to thank the referee for his valuable comments and suggestions.
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Received: June 09, 2015;Accepted: September 21, 2015
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